Dynamic Activity

Geometric Sequences

Lesson Vocabulary

• geometric sequence
• common ratio
• geometric mean

Take Note Key Concept: Geometric Sequence

A geometric sequence with a starting value a and a common ratio r is a sequence of the form

a , a r , a r 2 + a r 3 , … eh comma eh r comma eh , r squared , plus eh , r cubed , comma dot dot dot

A recursive definition for the sequence has two parts:

a 1 = a initial condition a n = a n − 1 ⋅ r , for   n > 1 recursive formula table with 2 rows and 3 columns , row1 column 1 , eh sub 1 , column 2 equals eh , column 3 initialcondition , row2 column 1 , eh sub n , column 2 equals . eh sub n minus 1 end sub . dot r comma for n greater than 1 , column 3 recursiveformula , end table

An explicit definition for this sequence is a single formula:

a n = a 1 · r n − 1 , for   n ≥ 1 eh sub n , equals , eh sub 1 , middle dot . r super n minus 1 end super . comma for n greater than or equal to 1

Problem 1 Identifying Geometric Sequences

Is the sequence geometric? If it is, what are a 1 eh sub 1  and r?

1. 3, 6, 12, 24, 48, …

   Think

   How do I find the ratios between consecutive terms?

   Divide the second term by the first term, then the third term by the second term, and so on.

   Find the ratios between consecutive terms.

   

   The common ratio is 2. The sequence is geometric with a 1 = 3 eh sub 1 , equals 3  and r = 2 . r equals 2 .